ART. 171] METHOD OF INVERSION. 85

169. If the given masses ml} m2, &c. are arranged so as to form an arc, surface or solid, the inverse masses will also be arranged in the same way. It will therefore be necessary to I discover some rule by which wT can compare the "density at any £

t of the given system with that at the corresponding point of [i ; the inverse system. |

': *"~*HDSng trie saine figure as before but changing the meaning of P, let PQ now represent any elementary arc of the locus of Q, then P'Q' represents the corresponding inverse arc. If the locus of Q is a curve, we infer from the similarity of the triangles POQ, P'OQ' that the lengths of the elementary arcs P'Q', PQ are in the ratio OQ70P, i.e. OQ'/OQ ultimately. Hence by (2) the ratio of the line densities of the arcs P'Q', PQ is equal to kjOQ'.

If the locus of Q is a surface, the elementary areas P'Q', PQ are in tEe ratio of the squares of the homologous sides, i.e. as OQ'2 to OQ2. Hence by (2) the ratio of the surface densities at Q' and Q is equal to (&/OQ')3.

~ If Q travel over all points of space enclosed by a surface, the elementary volumes at Q', Q are ultimately in the ratio OQ'2 dv.d(OQ') to OQ2da>.d(OQ). Since OQ.OQ' = &2, this ratio is equal to OQ'3/OQ3. Hence by (2) the ratio of the densities at Q' and Q is equal to (k/OQJ.

Summing these results, we see that

/ Density at Q' \ _ / density at Q \ / Ic Y^-1 }

\of the inverse system/ ~~ \of the given system/ ' \0 Q'J " '^ '' where d represents the dimensions of the system, i.e. d—1, 2, or 3 according as the system is an arc, a surface or a volume. When the system is a point, d = 0 ; the equation (4) then agrees with (2) y

and gives the relation between m and in'. X* ^

170. The mass of any portion of the inverse body is equal to ' * '

the potential at the centre of inversion of the corresponding portion of the primitive' body multiplied by the radius k of inversion. By Art. 168, we have m =mk/OQ, i.e. mf is equal to the potential of m at 0, multiplied by k. The theorem being true for each element of mass is necessarily true for any finite portion of the body.

171. Ex. If the law of force be the inverse ?ith power of the distance, the potential of a particle m takes the form — — -^L. Prove that the equations corresponding to (2), (3), and (4) become